Because the three parts of the Einstein equivalence principle
discussed above are so very different in their empirical
consequences, it is tempting to regard them as independent
theoretical principles. On the other hand, any complete and
self-consistent gravitation theory must possess sufficient
mathematical machinery to make predictions for the outcomes of
experiments that test each principle, and because there are
limits to the number of ways that gravitation can be meshed with
the special relativistic laws of physics, one might not be
surprised if there were theoretical connections between the three
sub-principles. For instance, the same mathematical formalism
that produces equations describing the free fall of a hydrogen
atom must also produce equations that determine the energy levels
of hydrogen in a gravitational field, and thereby the ticking
rate of a hydrogen maser clock. Hence a violation of EEP in the
fundamental machinery of a theory that manifests itself as a
violation of WEP might also be expected to show up as a violation
of local position invariance. Around 1960, Schiff conjectured
that this kind of connection was a necessary feature of any
self-consistent theory of gravity. More precisely, Schiff's
conjecture states that
any complete, self-consistent theory of gravity that embodies
WEP necessarily embodies EEP
. In other words, the validity of WEP alone guarantees the
validity of local Lorentz and position invariance, and thereby of
EEP.

If Schiff's conjecture is correct, then Eötvös experiments may
be seen as the direct empirical foundation for EEP, hence for the
interpretation of gravity as a curved-spacetime phenomenon. Of
course, a rigorous proof of such a conjecture is impossible
(indeed, some special counter-examples are known), yet a number
of powerful ``plausibility'' arguments can be formulated.

The most general and elegant of these arguments is based upon
the assumption of energy conservation. This assumption allows one
to perform very simple cyclic gedanken experiments in which the
energy at the end of the cycle must equal that at the beginning
of the cycle. This approach was pioneered by Dicke, Nordtvedt and
Haugan (see, e.g. [74]). A system in a quantum state
A
decays to state
B, emitting a quantum of frequency
. The quantum falls a height
H
in an external gravitational field and is shifted to frequency
, while the system in state
B
falls with acceleration
. At the bottom, state
A
is rebuilt out of state
B, the quantum of frequency
, and the kinetic energy
that state
B
has gained during its fall. The energy left over must be exactly
enough,
, to raise state
A
to its original location. (Here an assumption of local Lorentz
invariance permits the inertial masses
and
to be identified with the total energies of the bodies.) If
and
depend on that portion of the internal energy of the states that
was involved in the quantum transition from
A
to
B
according to

(violation of WEP), then by conservation of energy, there must
be a corresponding violation of LPI in the frequency shift of the
form (to lowest order in
)

Haugan generalized this approach to include violations of
LLI [74], (TEGP 2.5 [147]).

Box
. The
formalism

Coordinate system and conventions:
time coordinate associated with the static nature of the
static spherically symmetric (SSS) gravitational field;
isotropic quasi-Cartesian spatial coordinates; spatial vector
and gradient operations as in Cartesian space.

Matter and field variables:

rest mass of particle
a
.

charge of particle
a
.

world line of particle
a
.

coordinate velocity of particle
a
.

electromagnetic vector potential;

Gravitational potential:

Arbitrary functions:T
(U),
H
(U),
,
; EEP is satisfied if
for all
U
.

Action:

Non-Metric parameters:

where
and subscript ``0'' refers to a chosen point in space. If EEP
is satisfied,
.

The first successful attempt to prove Schiff's conjecture more
formally was made by Lightman and Lee [86]. They developed a framework called the
formalism that encompasses all metric theories of gravity and
many non-metric theories (Box
1). It restricts attention to the behavior of charged particles
(electromagnetic interactions only) in an external static
spherically symmetric (SSS) gravitational field, described by a
potential
U
. It characterizes the motion of the charged particles in the
external potential by two arbitrary functions
T
(U) and
H
(U), and characterizes the response of electromagnetic fields to
the external potential (gravitationally modified Maxwell
equations) by two functions
and
. The forms of
T,
H,
and
vary from theory to theory, but every metric theory satisfies

for all
U
. This consequence follows from the action of electrodynamics
with a ``minimal'' or metric coupling:

where the variables are defined in Box
1, and where
. By identifying
and
in a SSS field,
and
, one obtains Eq. (7). Conversely, every theory within this class that satisfies
Eq. (7) can have its electrodynamic equations cast into ``metric''
form. In a given non-metric theory, the functions
T,
H,
and
will depend in general on the full gravitational environment,
including the potential of the Earth, Sun and Galaxy, as well as
on cosmological boundary conditions. Which of these factors has
the most influence on a given experiment will depend on the
nature of the experiment.

Lightman and Lee then calculated explicitly the rate of fall
of a ``test'' body made up of interacting charged particles, and
found that the rate was independent of the internal
electromagnetic structure of the body (WEP) if and only if
Eq. (7) was satisfied. In other words, WEP
EEP and Schiff's conjecture was verified, at least within the
restrictions built into the formalism.

Certain combinations of the functions
T,
H,
and
reflect different aspects of EEP. For instance, position or
U
-dependence of either of the combinations
and
signals violations of LPI, the first combination playing the
role of the locally measured electric charge or fine structure
constant. The ``non-metric parameters''
and
(Box
1) are measures of such violations of EEP. Similarly, if the
parameter
is non-zero anywhere, then violations of LLI will occur. This
parameter is related to the difference between the speed of light
c, and the limiting speed of material test particles
, given by

In many applications, by suitable definition of units,
can be set equal to unity. If EEP is valid,
everywhere.

The rate of fall of a composite spherical test body of
electromagnetically interacting particles then has the form

where
and
are the electrostatic and magnetostatic binding energies of the
body, given by

where
,
, and the angle brackets denote an expectation value of the
enclosed operator for the system's internal state. Eötvös
experiments place limits on the WEP-violating terms in Eq. (11), and ultimately place limits on the non-metric parameters
and
. (We set
because of very tight constraints on it from tests of LLI.)
These limits are sufficiently tight to rule out a number of
non-metric theories of gravity thought previously to be viable
(TEGP 2.6 (f) [147]).

The
formalism also yields a gravitationally modified Dirac equation
that can be used to determine the gravitational redshift
experienced by a variety of atomic clocks. For the redshift
parameter
(Eq. (4)), the results are (TEGP 2.6 (c) [147]):

The redshift is the standard one
, independently of the nature of the clock if and only if
. Thus the Vessot-Levine rocket redshift experiment sets a limit
on the parameter combination
(Figure
3); the null-redshift experiment comparing hydrogen-maser and SCSO
clocks sets a limit on
. Alvarez and Mann [4,
3,
5,
6,
7] extended the
formalism to permit analysis of such effects as the Lamb shift,
anomalous magnetic moments and non-baryonic effects, and placed
interesting bounds on EEP violations.

The
formalism can also be applied to tests of local Lorentz
invariance, but in this context it can be simplified. Since most
such tests do not concern themselves with the spatial variation
of the functions
T,
H,
, and
, but rather with observations made in moving frames, we can
treat them as spatial constants. Then by rescaling the time and
space coordinates, the charges and the electromagnetic fields, we
can put the action in Box
1
into the form (TEGP 2.6 (a) [147])

where
. This amounts to using units in which the limiting speed
of massive test particles is unity, and the speed of light is
c
. If
, LLI is violated; furthermore, the form of the action above must
be assumed to be valid only in some preferred universal rest
frame. The natural candidate for such a frame is the rest frame
of the microwave background.

The electrodynamical equations which follow from Eq. (15) yield the behavior of rods and clocks, just as in the full
formalism. For example, the length of a rod moving through the
rest frame in a direction parallel to its length will be observed
by a rest observer to be contracted relative to an identical rod
perpendicular to the motion by a factor
. Notice that
c
does not appear in this expression. The energy and momentum of
an electromagnetically bound body which moves with velocity
relative to the rest frame are given by

where
,
is the sum of the particle rest masses,
is the electrostatic binding energy of the system (Eq. (12) with
), and

The electrodynamics given by Eq. (15) can also be quantized, so that we may treat the interaction of
photons with atoms via perturbation theory. The energy of a
photon is
times its frequency
, while its momentum is
. Using this approach, one finds that the difference in round
trip travel times of light along the two arms of the
interferometer in the Michelson-Morley experiment is given by
. The experimental null result then leads to the bound on
shown on Figure
2
. Similarly the anisotropy in energy levels is clearly
illustrated by the tensorial terms in Eqs. (16) and (18); by evaluating
for each nucleus in the various Hughes-Drever-type experiments
and comparing with the experimental limits on energy differences,
one obtains the extremely tight bounds also shown on Figure
2
.

The behavior of moving atomic clocks can also be analysed in
detail, and bounds on
can be placed using results from tests of time dilation and of
the propagation of light. In some cases, it is advantageous to
combine the
framework with a ``kinematical'' viewpoint that treats a general
class of boost transformations between moving frames. Such
kinematical approaches have been discussed by Robertson, Mansouri
and Sexl, and Will (see [144]).

For example, in the ``JPL'' experiment, in which the phases of
two hydrogen masers connected by a fiberoptic link were compared
as a function of the Earth's orientation, the predicted phase
difference as a function of direction is, to first order in
, the velocity of the Earth through the cosmic background,

where
,
is the maser frequency,
L
=21 km is the baseline, and where
and
are unit vectors along the direction of propagation of the light
at a given time and at the initial time of the experiment,
respectively. The observed limit on a diurnal variation in the
relative phase resulted in the bound
. Tighter bounds were obtained from a ``two-photon absorption''
(TPA) experiment, and a 1960s series of ``Mössbauer-rotor''
experiments, which tested the isotropy of time dilation between a
gamma ray emitter on the rim of a rotating disk and an absorber
placed at the center [144].